Flute Acoustics: an Introduction

Flute acoustics: an introduction How does a flute work? This introduction gives first the simple explanations, then the subtleties. It requires no mathematics, nor any special acoustics knowledge. Some more technical references are listed near the end of this page. Overview The flutist blows a rapid jet of air across the embouchure hole. The pressure inside the player's mouth is above atmospheric (typically 1 kPa: enough to support a 10 cm height difference in a water manometer). The work done to accelerate the air in this jet is the source of power input to the instrument. The player provides power continuously: in a useful analogy with electricity, it is like DC electrical power. Sound, however, requires an oscillating motion or air flow (like AC electricity). In the flute, the air jet, in cooperation with the resonances in the air in the instrument, produces an oscillating component of the flow. Once the air in the flute is vibrating, some of the energy is radiated as sound out of the end and any open holes. A much greater amount of energy is lost as a sort of friction (viscous loss) with the wall. In a sustained note, this energy is replaced by energy put in by the player. The column of air in the flute vibrates much more easily at some frequencies than at others (i.e. it resonates at certain frequencies). These resonances largely determine the playing frequency and thus the pitch, and the player in effect chooses the desired set of resonances by choosing a suitable combination of keys. In this essay, we look at these effects one by one. The air jet vibrates The jet of air from the player's lips travels across the embouchure-hole opening and strikes against the sharp further edge of the hole. If such a jet is disturbed, then a wave-like displacement travels along it and deflects it so that it may blow either into or out of the embouchure hole. The speed of this displacement wave on the jet is just about half the air-speed of the jet itself (which is typically in the range 20 to 60 metres per second, depending on the air pressure in the player's mouth). The origin of the disturbance of the jet is the sound vibration in the flute tube, which causes air to flow into and out of the embouchure hole. If the jet speed is carefully matched to the frequency of the note being played, then the jet will flow into and out of the embouchure hole at its further edge in just the right phase to reinforce the sound and cause the flute to produce a sustained note. To play a high note, the travel time of waves on the jet must be reduced to match the higher frequency, and this is done by increasing the blowing pressure (which increases the jet speed) and moving the lips forward to shorten the distance along the jet to the edge of the embouchure hole. These are the adjustments that you gradually learn to make automatically when playing the flute. Flutists are usually taught to reduce the lip aperture when playing high notes. The figure at left shows a jet striking an edge and being alternately deflected up and down. The sketch at right represents a cross section of the flute at the embouchure. The flute is an open pipe The flute is open at both ends. It's obvious that it's open at the far end. If you look closely at someone playing a flute, you'll see that, although player's lower lip covers part of the embouchure hole, s/he leaves a large part of the hole open to the atmosphere, as shown in the sketch above. Let's begin by considering a pipe that is simpler than a flute. First, we shall pretend that it is a simple cylindrical pipe---in other words we shall assume that all holes are closed (down to a certain point, at least), that the head is cylindrical, and we shall replace the side mounted embouchure hole with a hole at the end. In fact, this is more like a shakuhachi than a flute. It's a crude approximation, but it preserves much of the essential physics, and it's easier to discuss. (We shall introduce the effects of finger holes and the embouchure geometry below, or you can consult our research papers on this topic.) The animation below is from Open vs closed pipes (Flutes vs clarinets), which gives a more detailed explanation. It shows a shows a pulse of high pressure reflecting in a pipe open to the air at both ends. Note that a complete cycle of vibration is the time taken for the pulse to travel twice the length L of the flute (once in each direction). The pulse travels at the speed of sound v, so the cycle would repeat at a frequency of v/2L, as we shall see again below. The natural vibrations of the air in the flute are due to resonances. The reflecting pulse of air in the animation is an example of such a resonance, the fundamental or lowest resonance of the flute. There is more about resonances in the page on standing waves. What standing waves or resonances are possible in an open cylindrical tube? We shall now answer this question in terms of sine waves and harmonics. The fact that the pipe is open to the air at the ends means that the total pressure at the ends must be approximately atmospheric pressure or, in other words, the acoustic pressure (the variation in pressure due to sound waves) is zero. These points are called pressure nodes, and they effectively lie past the end of the tube by a small distance (about 0.6 times the radius, as shown: this distance is called the end correction). Inside the tube, the pressure need not be atmospheric, and indeed for the first resonance, the maximum variation in pressure (the pressure anti-node) occurs at the middle. The standing wave is sketched below. The bold line is the variation in pressure, and the fine line represents the variation in the displacement of the air molecules. The latter curve has anti-nodes at the ends: air molecules are free to move in and out at the open ends. (Note that a node for pressure and a node for air motion are not the same thing: indeed, pressure nodes often coincide with motion anti-nodes and vice versa. See pipes and resonances. The difference between closed and open pipes is explained in Open vs closed pipes (Flutes vs clarinets), which compares them using wave diagrams, air motion animations and frequency analysis, or some more flow animations.) The wave shown above is the longest standing wave that can satisfy this condition of zero pressure at either end. In the figure below, we see that it has a wavelength twice as long as the flute. The frequency f equals the wave speed v divided by the wavelength l, so this longest wave corresponds to the lowest note on the instrument: C4 on a C foot instrument. (Flutists please note: this page uses the standard note names, not the names sometimes used by flutists.) You might want to measure the length L of your flute, take the speed of sound as v = 350 metres per second for sound in warm, moist air, and calculate the expected frequency. Then check the answer in the note table. (You will find that the answer is only approximate, because of end corrections.) You can play C4 on the flute with this fingering, but you can also play other notes by blowing harder, or by narrowing the lip aperture (either gives a faster jet). These other notes correspond to the shorter wavelength standing waves that are possible, subject to the condition that the sound pressure be zero at both ends. The first several of these are shown in the diagram below. The series of notes with frequency fo, 2fo, 3fo etc is called the harmonic series, and notes with these frequencies have the pitches shown below. With all the tone holes closed, the first ten or so resonances of the flute are approximately in this ratio, so you can play the first seven or eight of the series by closing all the tone holes and blowing successively harder (or by narrowing the lip aperture). Note the half sharp on the seventh harmonic - it falls roughly midway between A6 and A#6. (You might be interested to compare this with the analogous diagram and sound files for the clarinet, which has only the odd harmonics present. There is also a more detailed discussion of the harmonic series ofopen and closed pipes.) Eight 'harmonics' of the lowest note on a flute. Each of the standing waves in the sketch above corresponds to a sine wave. The sound of the flute is a little like a sine wave (a very pure vibration) when played softly, but successively less like it as it is played louder. To make a repeated or periodic wave that is not a simple sine wave, one can add sine waves from the harmonic series. So C4 on the flute contains some vibration at C4 (let's call its frequency fo), some at C5 (2fo), some at G5 (3fo), some at C6 (4fo), etc. The 'recipe' of the sound in terms of its component frequencies is called its spectrum. (See sound spectrum for an explanation.) Looking at real sound spectra for played C4 (Open a new window for C4) you will see that, atpianissimo, the first harmonic (fundamental) and the frequency of the note C4 dominates, and that the higher harmonics become more important as the note is played more loudly, and as the flute develops a richer tone and sounds less and less like a sine wave.

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